When Bode Miller, the Olympic ski star known for daring Alpine racing, met Sara A. McKenna in San Diego last year through the high-end matchmaker Kelleher International, they were both professing interest in finding a marriage partner, she recalls.

The relationship did not last long — but she did become pregnant. And now the skier, 36, and Ms. McKenna, 27, a former Marine and firefighter who is attending Columbia University with G.I. Bill support, are locked in a cross-country custody fight that has become not only tabloid fodder but also a closely watched legal battle over the rights of pregnant women to travel and make life choices.